Prospects for healing the deep wound in the Republican Party left by the budget fight depend on whether President Bush overcomes counsels of caution and proposes a new tax-cutting initiative to fight the recession and reestablish his credibility.

Only a week ago, the president's two principal advisers, chief of staff John Sununu and budget director Richard Darman, were clearly opposed to reopening the tax question. Since then, pressure from Jack Kemp, William Bennett, Newt Gingrich, Vin Weber and junior White House policy staffers has forced serious consideration of the tax question.

But limitations imposed by politics and by the pay-as-you-go budget deal itself weigh heavily on the administration. Sununu and Darman, terrified of the "fairness" issue, are still loath to give Democrats a vehicle for income redistribution by proposing a capital gains tax cut. The answer pressed by the tax-cutters is that the Republicans must abandon caution and forthrightly face the recession with a package extending beyond capital gains to cuts in the Social Security payroll tax as well.

To flinch from taking that course threatens more upheaval within the party, including the possible resignation of Kemp as secretary of housing. Bennett would be placed in an embarrassing position as the new Republican national chairman. House GOP leaders Gingrich and Weber would promote their own tax-cut program, further alienating them from the White House.

What is at stake was dramatized Monday, when Darman spent two hours alone with Gingrich, who had called for him to resign, and Weber. Turning on the charm, Darman protested that his celebrated speech trashing Reaganism and Republican reforms had been badly misinterpreted. He was not a "window washer," he said, but an "architect" of reform. To prove it, he gave the two congressmen an autobiographical account of his exploits over the years.

They were unimpressed. They told Darman bluntly they held him responsible for betraying and undermining the values of Republicanism established over the past decade. Gingrich's test is simple. Since Darman's well-publicized speeches over the years have lobbed bombs at conservative institutions, he should deliver a new speech embracing the conservative reform he assured Gingrich and Weber that he supports.

The real test for both Darman and the Bush administration is what the president says about taxes in the State of the Union address. The clear inclination has been to call for "no more new taxes" and, apart from the fine print in the new budget proposal, shelve capital gains cuts and any other tax initiatives next year.

That resolve was shattered when Darman's anti-reform speech brought Kemp into open opposition. He argued that the president, facing a deepening recession with no program in hand, must play the tax-cut card. Bennett, picked by Sununu as party chairman to quiet the right, immediately joined Kemp.

Two Bush political operatives, Robert Teeter and Charles Black, were enlisted to bring Bennett back into line. Their argument is that the Democrats are poised to exploit the "fairness" issue by taxing millionaires and giving benefits to the poor. If Bush gives them a vehicle by proposing a capital gains cut, he could well end up vetoing his own bill -- an argument straight from Sununu.

That was not enough to turn Bennett. The new party boss does not disguise his ignorance of economics, but he knows better than the president's advisers that Republicans must return to their tax-cutting base. Even before his installation as party chief, he is sitting in on exploratory surgery sessions at the White House, analyzing the political pitfalls of tax reduction.

Quite apart from politics, pay-as-you-go provisions in the budget deal may preclude a capital gains tax reduction that, though actually producing additional revenue, would be "scored" by the Democratic-controlled congressional budget offices as losing money. Thus does the budget deal, source of so much discord in the party, continue to confound Republican unity.

It is fair to say that if the current political mind-set and the budget deal had been in effect in 1981, no Reagan tax cut would have even been proposed, much less enacted. Sununu is being challenged to stop worrying about the "fairness" issue and override the bean counters by advising the president to fight the recession with lower taxes. Which way he goes may determine the shape of politics and the economy far into the future.